Abstract
Thank you for looking at the report for my simulation code sample in
R! This is a report for a project I thought of when I came
across while doing a questionnaire for the World Bank’s Development
Impact Evaluation (DIME). The question was along the lines of:
“There is a program that is implemented at the village level.
Households within the same village are very similar but households
between villages are not. To maximize the likelihood of detecting the
programs effect is it better to sample more households within each
village or to sample more villages?”
In this report I will answer this question through the use of
simulation techniques in which programs with different effect sizes are
implemented on a sample of randomly selected villages. I will also go
over some of the theory and intuition for the answer and take this
opportunity to talk about some sampling techniques, mainly on clustered
sampling vs. stratified sampling vs. systematic sampling. However, the
main objective of this report is to demonstrate my skills in
R as such this document will mainly focus on the
code itself.
for any given problem there are many different solutions and paths. While some paths may be more efficient or shorter than others. There are two necessary conditions for good code. Good code must be:
One can not come at the expense of the other. These principles are behind every decision and form the backbone of every script in every language. I am confident you will see this reflected in my work.
One of the most important ways to create good code is to follow good coding practices from the beginning, as going back to fix things will always be costlier than starting out the right way. Throughout this report you will see blue text boxes that explain some of the styling decisions made throughout the script to guarantee the functionality and readability.
This is an example of a style textbox.Everything in this report is my own work with all relevant citations
included. However, you will find that plural first person active voice
is often used throughout. This is a deliberate choice. In these cases
“we” refers to me (the author of the report) and you (the reader). This
is meant to make the report more engaging and less formal. This is not a
formal writing sample, as the main purpose of this document is to
showcase my programming abilities and experience in R. If
you wish to see a formal or an academic writing sample please write me an email.
Making sure your code is reproducible and portable is also essential
for good code. I always create a new R-project for each assignment,
maintain Renvironments through renv and
detailed records of every change through version control (as you can
probably tell by reading this document on GitHub). In fact, this project
not only has renv to increase its reproducibility and
portability it also contains a mamba directory with the
.Rprofile and config.yml files needed to
guarantee that no matter when or in what system, the process and results
can be replicated. This means that this project is 100%
reproducible and portable. Just remember not to use
mamaba and renv simultaneously as they can
conflict with each other.
I believe feedback is incredibly valuable. Because of this, if you have any ideas, comments, criticisms, questions, advice, or if you find a mistakes or are having trouble replicating the results please let me know. After reading this report, I would highly appreciate it if you could complete my survey on the general perceptions of my dossier. Likewise, if you have any specific inquiries feel free to contact me though my email ja.ortiz@uniandes.edu.co.
This is a question I came across while completing a questionnaire for the World Bank’s Development and Impact Evaluation (DIME). While not a verbatum quote, the question was:
There is a program that is implemented at the village level. Households within the same village are very similar but households between villages are not. To maximize the likelihood of detecting the programs effect is it better to sample more households within each village or to sample more villages?
Intuitively one may think that it s better to sample more villages. If households within each village are similar then the information that an additional household from a village that has already been sampled contributes to the regression is than the information contributed from a household from village that is unsampled and which there for, is different to all the other households in the sample.
As mentioned above, intuitively one might expect that sampling households from different villages would increase the statistical significance of the estimator. Lets take a look at the first graph. It’s worth saying that all these graphs are interactive yo you may pan, rotate, zoom, etc. as well as hover over the plot to see the number of villages per each treatment group (i.e. treated and control), total sample size and the p-vale with ‘*’ at each of the usual significance thresholds (10%, 5%, 1%).
From this graph it is not immediately obvious that either sampling strategy is better than the other. In fact, it seems as if the surface descends at the same rate regardless if you are increasing the number of households per village or if you are increasing the number of villages per treatment group. Additionally, the surface of this graph is very rough, there are many local maxima and local minima scattered throughout. This is of course expected as a result from idiosyncratic errors and sample variance. However it is nevertheless surprising, as this graph shows the average p-value over 1000x runs.
Let’s now see how this graph changes as we increase the effect size. Once again I encourage you to explore each graph.
From the graphs above, it is easier to see that a pattern starts to emerge. Perhaps the most noticeable thing is how quickly the effects become significant (thus far we haven’t looked at the estimators. For now, just know that in this context they are consistent and unbiased). The second most noticeable thing is how much smoother the surface is. In part this is due to the zero lower bound on p-values and in part it is the result of the surface ‘stretching’ as p-values decrease faster the larger the effect is.
Thirdly, and perhaps most importantly, by playing around with the graphs one might find that increasing the number of villages per treatment group is more effective at reducing the p-value than increasing the number of households per village.
By hoovering your mouse over the graphs you’ll see that for any given sample size, the p-value is lower in cases where there are more villages per group than households per village. That is to say that if you looked at the surface from above so that you only saw the x (Households per village) and y (Villages per treatment group) axis, and drew a 45° line on \(y = x\), the values above the line (i.e. more villages than households) will in average be lower than values below the line (i.e. more households than villages). In other words, the surface is slanted so that increasing the number of villages will lower the p-value more than increasing the number of households. The clearest sign of this can be seen on the slanted border on the right side of the initial view (i.e. \(Households\ per \ village = 1\)). the surface limit reduces as the number of villages increases whereas on the opposite border,(i.e. \(Villages\ per\ treatment\ group = 1\)) the surface value remains at the same level regardless if the number of households sampled per village is 50 or 2.
After looking at the results we now turn our attention to process. This is the part where we take an in-depth look at the code.
The script starts with some metadata. This is the scripts title, the author and a brief description of what this script does. In this case the script also includes a warning to look at this report before looking at the script itself.
# Code sample R
# By: Alejandro Ortiz - ja.ortiz@uniandes.edu.co
# This is the an exercise that models a question from DIME.
# IMPORATANT !!
# Please look at the HTML report before looking at this scriptNext comes the Set up section. The section begins with a
series of commands to clean R’s environment by clearing the
console, the ploting device, as well as any environment variables or
objects and preforming some memory clean up. Here, the working directory
is also printed to console (in the past this has spared me form a
lot of confusion when opening a script and not realizing the
project in which it its opened). There is also a setting to change the
maximum console output to 200 (don from the default of 1000) as I find
200 is more that enough for any type of work limiting this setting is
important because helps to keep track of executions, maintain order in
the console, and helps you better understand console of outputs.
Finally, the relevant packages are loaded they are grouped according to
their function and the most important packages are
always placed last to prevent masking by other
packages. As you can see the most important packages are
tydiverse and data.table , two amazing pieces
of software that make R what it is today.
# Set up ----
# Clean - R's environment
# .rs.restartR()
cat("\f")
# dev.off()
remove(list = ls()); gc(full = T)
# Publish working directory
getwd()
# Set options
# options(java.parameters = "-Xmx8000m")
options(max.print = 200)
# Update and load packages
# update.packages(ask = F)
# Plot results
library(plotly)
# Paralelization
library(furrr)
# Estimation
library(fixest)
# Core
library(tidyverse)
library(data.table)options(java.parameters = "-Xmx8000m")
Is useful when working with rJava, even though it is not used here I keep it in the set up as a reminder that Java can run into memory problems if this setting is not enabled. This way if in the future any Java is used this option can be easily enabled.The question is quite broad, and hence it’s abstracted from a lot of
the details needed for the simulation. These details are all compiled in
the next section titled Hyper-parameter dashboard. In here
it’s possible to easily change all of these hyper-parameters such as the
minimum and the maximum number of households per village. This section
also includes some options used for exporting the results. However,
looking at these parameters without knowing what context they are used
in isn’t very useful. So I wont go into much detail here, instead I’ll
simply say that all these user-defined values are stored in a list of
two elements. The first one is the hyper-parameters, i.e. things that
directly influence the outcome of the simulation. The second is the
options, i.e. things that affect how the results are presented but leave
the actual values the same.
# Hyper-parameter dashboard ----
# Hyper-parameters list template
l <- list()
# Maximum number of households sampled per village
l$params$max_hh_sample_per_vil <- 50
# Maximum number of villages sampled per treatment group
l$params$max_vil_sampled <- 50
# Minimum effect size
l$params$min_effect_size <- 0.1
# Maximum effect size
l$params$max_effect_size <- 0.3
# Effect size step
l$params$effect_size_step <- 0.05
# Number of times to run simulation over the same sample parameters
l$params$nu_simulations <- 10^4
# Minimum number of villages in the population
l$params$min_vils_in_universe <- 200
# Minimum number of HH per village in population
l$params$min_vil_size <- 50
# Maximum number of HH per village in population
l$params$max_vil_size <- 200
# Independent probability of a village being treated
l$params$prob_vil_is_treated <- 0.5
# Minimum value of the mean of baseline score
l$params$bl_min_mean_val <- 2
# Maximum value of the mean of baseline score
l$params$bl_max_mean_val <- 100
# Minimum value of the sd of baseline score
l$params$bl_min_sd_val <- 0.5
# Maximum value of the sd of baseline score
l$params$bl_max_sd_val <- 1.5
## Not Simulation parameters but options
# Save full list of hyper-parameters in file name?
l$opts$full_params_in_file <- F
# Select parameters to include in file name in case
# l$params$full_params_in_file == F
l$opts$selct_params_in_file <- list("eff" = quote(eff_size))Hyper parameters are placed on a list so they are 1) easy to acess while 2) ony taking up 1 slot in the global environementIn the future, warning and error handling will be applied to make sure that all values are consistent with each other (to prevent things like the maximum value being lower than the minimum).
The third code section is titled Simulations this is
where the actual simulations take place. I highly recommend you
change the hyper parameters before you attempt to run this
section as the default values can be quite onerous on your
system. It took me around one day to execute this part alone. This, in
spite of the code being fully parallelelized and achieving a consistent
CPU utilization of over 99% on all cores.
This section begins by defining the gen_village() and
change_hh_size() functions, much like with the hyper
parameters, looking at this functions without knowing what contexts they
are used in isn’t very informative so we will skip them for now with the
promise to dive in deeper into their process once they are called.
There are two other things at the beginning of this section that
are worth remaking upon. The first one is the the
paralellization plan (in this case using furrr’s
multisesion). The second is the beginning of a loop through
different effect sizes:
# plan for future processes
plan(multisession)
# Loop over effect size
for (
eff_size in seq(
from = l$params$min_effect_size,
to = l$params$max_effect_size,
by = l$params$effect_size_step
)
) { ...As you can see the loop will iterate over a list that starts at the
minimum effect size min_effect_size through the maximum
effect size max_effect_size in specified valued increments
of effect_size_step (as a reminder, all objects prefixed by
l$params$ are user defined hyper parameters).
The loop starts off by creating a template in which results will be
stored. This is a matrix titled m_pvals (m for
matrix and pvals since it will store p-values):
...
# Pre-allocate memory to results
m_pvals <- matrix(
nrow = l$params$max_vil_sampled,
# row number is equal to number of villager per treatment group
ncol = l$params$max_hh_sample_per_vil)
# column number is equal to number of HHs per village
# Matrix must not be NA or operations will only yield NA
m_pvals[is.na(m_pvals)] <- 0
...As noted in the comments, in this matrix columns represent the number of households sampled per village and rows represent the number of villages per treatment group. This is to say that the value \(m\_pvals_{ij}\) represents the p-value for a sample of \(i\) villages in both the treatment and control groups with \(j\) households in each village where the total sample size is \(i \times j \times 2\) (2 because there are two treatment groups).
These matrix is then filled with 0’s. This is done so that operations
(like addition) don’t result in NAs.
After this, a nested loop starts, this loop will iterate the
simulation experiment nu_simulations (1000x by default) and
average the results at the end:
...
# Simulate results many times
for (iter in 1:l$params$nu_simulations) {
# Set a fixed seed for reproducibility
set.seed(1944 + iter) # Year of Bretton woods
# Template of village-household universe
vil_hh_u <- data.table()
# Creating the universe - loop over villages
for (village in 1:l$params$min_vils_in_universe) {
vil_hh_u <- rbind(vil_hh_u, gen_village())
}
...To guarantee reproducibility since this loop will use quasi-random number generators, the seed is set to an arbitrary value that changes with every iteration. In this case since the question came from DIME, it seemed fitting to set the seed’s fixed component to the year of the Bretton Woods conference that resulted in the modern international economic system.
Then there is another nested loop, this loop is in charge of
generating the universe of all villages and households, so it runs
min_vils_in_universe (default: 200) times, once for each
village. On each iteration it concatenates the result of the new village
with all the previous results.
Each village is created using the gen_village() function
we see here:
# Function to generate a random village
gen_village <- function() {
# Select a village size
vil_size <- sample(l$params$min_vil_size:l$params$max_vil_size, size = 1)
# Select if village will be treated or not
treatment <- sample(0:1,
size = 1,
prob = c(1 - l$params$prob_vil_is_treated,
l$params$prob_vil_is_treated)
)
# Create a village data set
tmp_vil <- data.table(
# Village ID
"village" = village,
# Within-village Household ID
"household" = 1:vil_size,
# Create baseline score
"baseline" = rnorm(vil_size,
# Mean depends on the village
mean = runif(1,
min = l$params$bl_min_mean_val,
max = l$params$bl_max_mean_val),
# SD depends on village
sd = runif(1,
min = l$params$bl_min_sd_val,
max = l$params$bl_max_sd_val)),
# Treatment status - village-wide effect
"treatead" = treatment %>% rep(vil_size))
# Concatenate village data set with all previous villages
return(tmp_vil)
}This function starts by randomly selecting the village size (from a
uniform distribution) that is between the minimum and maximum village
size (min_vil_size, max_vil_size
respectively). It then assigns with probability
prob_vil_is_treated whether the village will be treated or
not. With this information it genrates a database that includes, the
village ID, the household ID (which is village-specific), a baseline
value for each household, and a dummy variable indicating if the village
was treated.
The baseline value is not associated to any particular characteristic. Likewise, the treatment is also completely abstract, this is becuase the original question makes no mention as to what type of outcome or treatment the program was, only that it was implemented at the village level.
The baseline value for each household comes form a normal distribution. The mean and standard deviation of this distribution are chozen at random. The choise of mean and SD comes from a uniform distribution with minimum and maximum values set by their own hyper-parameters. Importantly every village has a different mean and SD, but all households in the same village are sampled from the same distribution. This is what makes villages ‘different’ while households within each village are ‘similar’. Importantly, the default size of the SD, the possible values of the mean, and their relation to the effect size has been calibrated to make sure this condition holds; the mean has a very large range of possible values of \([2, 100]\), SD has a comparetivley small range of smaller vlaues: \([0.5, 1.5]\) and effect size ranges from \(0.1\) to \(0.3\).
After iterating over as many villages as the minimum number of villages specified and generating the data set the loop continues to check if there are sufficient villages in each treatment group to continue execution:
...
## Contingency in case treatment or control groups are too small
# Size of the smallest treatment group
smallest_group <-
vil_hh_u[, .(nu_vils = uniqueN(.SD, by = "village")), by = treatead
][, min(nu_vils)]
# Guarantee that there are enough villages in both treatment groups
while (smallest_group < l$params$max_vil_sampled) {
vil_hh_u <- rbind(vil_hh_u, gen_village())
# Size of the smallest treatment group
smallest_group <-
vil_hh_u[, .(nu_vils = uniqueN(.SD, by = "village")), by = treatead
][, min(nu_vils)]
}
...If there anen’t, it will continue to generate villages using
gen_village() and adding them to the universe of villages
vil_hh_u until there are enough villages in each treatment
group as the maximum number of villages sampled
max_vil_sampled. This gurantees that there are as many
villages per treatment group as rows in the m_pvals matrix,
but is only a contingency where the ratio of minimum number of villages
in the universe to the maximum number of villages per treatment group is
close to 1 (i.e. \(min\_vils\_in\_universe /
max\_vil\_sampled \approx 1\)).
This function serves three main purposes. First, it prevents code repetition; second, it guarantees that the underlying process for generating the data is the same no matter if its generated inside the initial for loop or the contingent while loop; third, by leverageing the previous two caracteristics, it improves the readability of the code.
Similarly, this function doesnt take any arguments because it doesnt
need to. The function does relly heavily on evioronment parametesr like
min_vil_size, prob_vil_is_treated, bl_min_mean_val, bl_min_sd_val
among others, but this functions is only designed to operate within the
scope of this script and is not meant to be generally applicable to any
R environment. So while it is possible to add arguments for parameters
like minim village size and the probability of a villaeg beeing treated
(amongs others), doing so would only make the code longer. Because
parcimony is a virtue I decided the additinal complexity would not
significatly contribute to the codes main objectives of function and
readability.
Once the universe of households and villages is created the outcome is generated for each household in each village. This is done by replicating the baseline value plus an additional random error term distributed \(N(\mu = 0, \sigma^2 = 1)\).
For treated units there is a homogenieus effect distributed \(N(\mu = \tau, \sigma^2 = 1)\) where \(\tau\) is the given effect size. As noted, this is a homogenieus effect, so all households in all villages are equallya fected by the treatment. Additionally, the treatment effect has a standard deviation of 1, a value that doesnt lead to any practical differences as the magnitude of the treatment is already changeing. As always, please do let me know if you wish to see any aditions to this simulation (like heterogeneus treatment effects).
...
# Homogeneous effects of treatment
vil_hh_u[, outcome := baseline + rnorm(.N)]
vil_hh_u[treatead == 1, outcome := baseline + rnorm(.N, mean = eff_size)]
# Treatment magnitude is the mean value
# New seed for stage-specific reproducibility
set.seed(2005 + iter) # Year DIME was created
## Parallelize
# Runs for different village sample sizes the effect of varying
# household sample size
p_result <- future_map(1:l$params$max_vil_sampled,
.f = change_hh_size,
universe = vil_hh_u,
.progress = T,
.options = furrr_options(seed = T))
...After the outcome value has been generated, a new seed is set. There is no need to do this as the first seed allready guaranteed the reproducibility of this section, too. Never the less it is placed here for practical purposes in case one only needs to replicate the estimation part of the simulation. Placing a new seed here means there is no deed to re-do the village/household generation in order to obtain the same results for estimation section.
The estimation itself is done using furrr’s
paralelization. for this the future_map() function is
called. Importantly for this fucntition the seed = T option
is set which makes sure that paralelization is done while respecting the
seed set above.
For the paralelization the change_hh_size() function is
called and paralelized through village sizez from 1 to the maximum
village sample size set by max_vil_sampled. Here, the
universe of villages & households vil_hh_u is passed on
as the universe argument.
The change_hh_size() takes the number of villages to be
sampled per treatment group nu_of_villages and a data set
of the universe of villages and households universe. It
then estimates the effect of varying the number of households sampled
per village and returns a vector of the p-values of each estimation.
Optionally one can also specify the maximum number of households sampled
per village (default: max_hh_sample_per_vil), and wheter or
not to verbose messages and warnigs from the estimation. The function
then returns is a 1xmax_hh_sample_per_vil (default: \(1 \times 50\)) vector of the p-values of
each estimation.
NOTE: This function could easily be exported to other environments if
the max_nu_hh_per_vil value is specified and the structure
of universe remains the same. Unlike with the
gen_village() function, his is important so
future can transport the function and it’s arguments into
another R session. However, this function is only designed
to be called in this scope. The universe and
max_nu_hh_per_vil arguments are included here to illustrate
the use of optional and mandatory parameters alongside more general form
of function writing.
change_hh_size <-
function(nu_of_villages, universe,
max_nu_hh_per_vil = l$params$max_hh_sample_per_vil,
return_messages = F, return_warnings = F
) {
# Required packages for the function
require(data.table)
require(fixest)
# Pre-allocate space for results into memory
pval_results <- vector("numeric", length = max_nu_hh_per_vil)
# Loop over HHs sampled in each village
for (nu_hh in 1:max_nu_hh_per_vil) {
...When the function is initiated the required packages are attached and a vector template for the results is created. Templates are important becase they pre-allocate the memory space for results which speeds up code as the computer only needs to re-write preexisting values when saving results.
After this, the loop over the number of households sampled per
village is initiated. The loop begins by creating a list of randomly
sampled treated and untreated villages. Before creating a temporary
template vector of lenght 1, which is important so future
can transport this object to other R sessions.
...
# Sample of treated villages
v_sample <- universe[treatead == 1, unique(.SD), .SDcols = "village"
][sample(.N, size = nu_of_villages)]
# Sample of untreated villages
v_sample <- rbind(
v_sample,
universe[treatead != 1, unique(.SD), .SDcols = "village"
][sample(.N, size = nu_of_villages)])
# Temporary result template
tmp <- 0
# Used so furrr - can transport the object
...Once the village sample is determined, estimation is preformed over
these villages. Inside each village a random sample of households is
drawn. The combination of the randomly drawn villages and randomly drawn
households is what makes the sample for a simple OLS estimation. In this
case the feols() fucntion from the fixest
package is used because this is an incredible package that is much more
feature-rich and much faster that the base::lm()
function. For the most part, niether of these advantages are needed in
this particular case, however, I heavily favour using
feols() over lm() even in cases where the
result is simillar.
...
# From a simple OLS - get the p-value for treated dummy
tryCatch(
expr = {
tmp <<-
feols(outcome ~ baseline + treatead,
data = universe[
# Only villages in sample
v_sample, on = "village"
# Sample a of hh in each village
][, .SD[sample(.N, size = min(.N, nu_hh))],
by = village]
# Grab p-value for treated dummy
)[["coeftable"]][["Pr(>|t|)"]][3]
},
warning = function(war) {
# Return warning if necessary
if (return_warnings) {
war
}
},
message = function(mes) {
# Return message if necessary
if (return_messages) {
mes
}
}
)
# Save to result vector
pval_results[nu_hh] <- tmp
}
# Result is the vector of p-values - for that Nu. of villages in each
# treatment group
return(pval_results)
}You will also notice that the estimation is wrapped in the
tryCatch() function. This is to include some basic form of
error/warning handling in the function. Usually when sample sizes are
small (i.e. when there is only a few households and only a few villages)
feols() might give a warning or remove a variable because
of colinearity, resulting in messages and warnings that can clutter the
console. To avoid this, wanings and messages are catched but can be
optionally returned using the return_messages and
return_warnings arguments.
Estimation results are then saved into the pval_results
vector and the vector is returned once the loop over all household
sample sizes is finished.
Results from future_map() are in list form (one could
use other map variants). They are then unlisted and placed
in a matrix row-wise, which is important for cases where the results
matrix isn’t symetrical.
If there are any NA values within the estimation results a warning is issued with the effect size and iteration where the NAs where detected. this is important because if there are a lot of NAs the calculated average of p-values could be incorrect. With the default det of hyper-parameters there are no NAs.
In case there are any NAs they are converted into 1s to avoid loosing
all the p-values from past and future iterations. After this, the matrix
with the p-values from that specific itetration tmp_mat is
added to all the other p-values from previous iterations contained in
m_pvals. The value of 1 is chosen to bias the results
upwards in case there is an NA.
Once an iteration is done, a message stating the iteration, effect size and the time is prited to console.
...
# Unlist results into a matrix
tmp_mat <- p_result %>% unlist %>%
matrix(ncol = l$params$max_hh_sample_per_vil, byrow = T)
# columns represent number HHs sampled per village,
# rows represent the number of villages per treatment group
# Warn if there are NAs, then replace NA's with 0
if (any(is.na(tmp_mat))) {
warning(
paste0("NAs detected for effect size: ", eff_size,
" Iteration: ", iter)
)
}
tmp_mat[is.na(tmp_mat)] <- 1
# Add results from previous iterations
m_pvals <- m_pvals + tmp_mat
# Verbalize iteration and time
cat(
paste0(
"Finished cycle: ", iter, " - for effect size: ", eff_size,
"\n", Sys.time(), "\n\n")
)
flush.console()
}
...After all iterations are done, the average of all p-values is taken by dividing the sum of all p-values across iterations by the number of iterations.
The case where there are only two observations, one treated and one control, is changed so its value is NA. This is because there are not enough data poins in the regressions which results in incorrectly estimated p-values.
Lastly, the results are saved to disk using the user specified options before moving onto the next iteration of the effect size loop.
Once all iteratios of the effect size loop are finished
future’s plan is set back to the default and the
environment is cleand of all non-hyper parameters.
...
# Calculate average p-value
avg_pvals <- m_pvals / l$params$nu_simulations
# Sample size of 2 is too small
avg_pvals[1, 1] <- NA
# Save full list of hyper parameters?
if (l$opts$full_params_in_file) {
file_details <-
paste(
paste0(names(l), "_", l),
collapse = " ")
} else {
file_details <-
paste(
paste0(names(l$opts$selct_params_in_file),
"_",
# Evaluate expressions in current setting
lapply(l$opts$selct_params_in_file, eval)),
collapse = " ")
}
# Transform to data.table
avg_pvals <- avg_pvals %>% as.data.table
# Bulk rename
names(avg_pvals) <- paste("HH_per_vil", 1:l$params$max_hh_sample_per_vil,
sep = ".")
# Add variable indicating the number of villages
avg_pvals[, nu_villages := 1:.N]
# Save result
fwrite(avg_pvals,
paste0("Outputs/HH - Village surface/",
"Homogeneous effects ",
file_details,
".csv"),
yaml = T)
}
# End parallelization
plan(sequential)
# Clean up after loop
remove(list = ls()[!ls() == "l"])fin